Decision Mathematics 2: Decision analysisEdexcel A-Level Further Maths: Topic test

1. 1
   A street-food trader must choose one of two pitches for the weekend. Pitch $\mathrm{P}$ gives a profit of £400 with probability $0.6$ and a profit of £100 with probability $0.4$. Pitch $\mathrm{Q}$ gives a certain profit of £250.

   (a)

   What is the expected monetary value (EMV) of pitch $\mathrm{P}$?

   [1 mark]

   • A£250
   • B£280
   • C£220
   • D£500

   (b)

   The trader chooses the pitch with the greater EMV. Which pitch is chosen, and by how much is its EMV greater?

   [1 mark]

   • APitch $\mathrm{Q}$, by £30
   • BPitch $\mathrm{P}$, by £150
   • CPitch $\mathrm{Q}$, by £150
   • DPitch $\mathrm{P}$, by £30

   (c)

   The certain profit from pitch $\mathrm{Q}$ is changed to £$x$, where $x$ is a whole number. Find the least value of $x$ for which the trader, choosing by EMV, would choose pitch $\mathrm{Q}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A theatre company can stage a risky play or a safe play. The risky play gives a profit of £10 000 with probability $0.5$ and a loss of £2 000 with probability $0.5$. The safe play gives a certain profit of £3 000. The company assigns utility $0$ to a loss of £2 000, utility $1$ to a profit of £10 000 and utility $0.55$ to a profit of £3 000.

   (a)

   What is the EMV of the risky play?

   [1 mark]

   • A£4 000
   • B£5 000
   • C£6 000
   • D£8 000

   (b)

   What is the expected utility of the risky play?

   [1 mark]

   • A$0.55$
   • B$1$
   • C$0.5$
   • D$0.25$

   (c)

   State which play the company should choose if it maximises expected utility, and explain what this suggests about its attitude to risk.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A developer is deciding whether to build a wind farm on a site. If it builds, the farm makes a profit of £6 million if the site is windy and a loss of £3 million if it is not. Without further information, the probability that the site is windy is $0.5$. Alternatively, the developer can first pay £0.4 million for a survey. The survey is favourable with probability $0.5$. If it is favourable, the probability that the site is windy is $0.8$; if it is unfavourable, that probability is $0.2$. After the survey the developer chooses whether or not to build; not building gives a profit of £0 (before the cost of the survey is deducted).

   (a)

   Calculate the EMV, in £ million, of building without a survey, and the EMV of building after a favourable survey result (before the cost of the survey is deducted).

   [3 marks]

   (b)

   Determine whether the developer should commission the survey, and state the optimal strategy.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A company has a patent. It can either license the patent to another firm for a certain £0.9 million, or develop a prototype itself at a cost of £0.5 million. The prototype works with probability $0.6$. If it does not work, the company receives nothing more. If it works, the company chooses between launching the product and selling the design for £2 million. A launch gives revenue of £5 million with probability $0.7$ (high demand) and £1 million with probability $0.3$ (low demand). The prototype cost is not included in these revenue figures.

   (a)

   Use EMVs to determine the company's best strategy, and state its EMV.

   [6 marks]

   (b)

   The board uses expected utility instead. For net profit in £ million, it assigns utility $0$ to $-0.5$, $0.4$ to $0.5$, $0.55$ to $0.9$, $0.7$ to $1.5$ and $1$ to $4.5$. Find the strategy that maximises expected utility, and comment on the board's attitude to risk.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A coach company must buy either a 49-seater coach or a 29-seater coach. Demand will be high with probability $0.4$ and low with probability $0.6$. If demand is high, the 49-seater earns an annual profit of £90 000 and the 29-seater earns £50 000. If demand is low, the 49-seater earns £10 000 and the 29-seater earns £30 000.

   (a)

   What is the EMV of the 49-seater coach?

   [1 mark]

   • A£58 000
   • B£100 000
   • C£38 000
   • D£42 000

   (b)

   Suppose the probability of high demand is $h$. The 49-seater has the greater EMV only when $h$ is greater than which value?

   [1 mark]

   • A$\frac14$
   • B$\frac13$
   • C$\frac12$
   • D$\frac23$

   (c)

   The owner is risk averse. Explain, with reference to the profits, which coach the owner is likely to prefer.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   An option has three possible outcomes: a profit of £40 000, a profit of £10 000 and a loss of £20 000. Their probabilities are $0.2$, $0.5$ and $x$ respectively. A certain alternative gives a profit of £6 000. A decision-maker's utilities are $0.6$ for £40 000, $0.45$ for £10 000, $0.4$ for £6 000 and $0$ for a loss of £20 000.

   (a)

   What is the value of $x$?

   [1 mark]

   • A$0.7$
   • B$0.8$
   • C$0.3$
   • D$0.5$

   (b)

   What is the EMV of the option?

   [1 mark]

   • A£7 000
   • B£19 000
   • C£30 000
   • D£6 000

   (c)

   Use expected utility to decide between the option and the certain alternative.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A biotech start-up can pay £1 million for a trial of a new compound, or sell the compound immediately for £1.2 million. The trial succeeds with probability $0.5$. If it succeeds, the start-up chooses between two options: it can license the compound for a certain net profit of £4.5 million, or manufacture it itself, which gives a net profit of £8 million if demand is high (probability $0.6$) and £1 million if demand is low (probability $0.4$). If the trial fails, the start-up receives nothing further. These net profits do not include the cost of the trial.

   (a)

   If the trial succeeds, determine which option the start-up should choose, showing the EMV of manufacturing.

   [3 marks]

   (b)

   Find the EMV of running the trial, and state whether the start-up should run the trial or sell the compound immediately.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   An investor must choose one of three investments. A bond gives a certain profit of £4 000. A property fund gives a profit of £15 000 with probability $0.5$ and a profit of £2 000 with probability $0.5$. A share portfolio gives a profit of £30 000 with probability $0.3$, a profit of £5 000 with probability $0.4$ and a loss of £10 000 with probability $0.3$.

   (a)

   Calculate the EMV of each investment and state which the investor should choose if maximising EMV.

   [6 marks]

   (b)

   The investor's utilities are $0$ for a loss of £10 000, $0.5$ for £2 000, $0.7$ for £4 000, $0.72$ for £5 000, $0.85$ for £15 000 and $1$ for £30 000. Use expected utility to decide which investment to choose, and comment on the difference from your answer to (a).

   [6 marks]

   Total for question 8: 12 marks

End of questions